Derivations, gradings, actions of algebraic groups, and codimension growth of polynomial identities

Suppose a finite dimensional semisimple Lie algebra g acts by derivations on a finite dimensional associative or Lie algebra A over a field of characteristic 0. We prove the g-invariant analogs of Wedderburn - Mal'cev and Levi theorems, and the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of A. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur's conjecture for finite dimensional associative algebras with an action of a reductive affine algebraic group by automorphisms and anti-automorphisms or graded by an arbitrary Abelian group not necessarily finite. In addition, we provide criteria for G-, H- and graded simplicity in terms of codimensions.